Floer Homology Language
TANAKA Akio
Note 3
Grothendieck Group
1
Topological space X
All the sets of continuous families' isomorphic classes ofZ2 graded Hermitian vector bundle
over X KH(X)
Set thatKH (X) is divided by equivalence relation as homotopy K(X)
K(X) is called X's K-group.
2
Compact Haussdorff space X
Set of all the isomorphic classes of complex vector bundle over X Vect(X)
Vect(X) has structure of commutative semigroup by direct sum.
ΦX : Vect(X) →K(X)
3
(Definition : Grothendieck Group)
Commutative semigroup S
Commutative group AS
Homomorphism as semigroup ΦA : S →AS
Arbitrary module R
Arbitrary homomorphism ΦR : S →R
Homomorphism Ψ : AS→R (1)
ΦR = ΨΦs (2)
(1) and (2) are uniquely existent.
4
If there exists Grothendieck group, it is unique.
5
(Theorem)
When X is compact Hausdorff space, (K(X), ΦX ) is Vect(X)'s Grothendieck group.
6
(Identification)
Language is identified with Vect(X).
Language is commutative by associative law.
Word is identified with (K(X), ΦX ).
Distance of word is identified with Hermitian vector bundle.
<Addition>
7
Element of KH(X) F
Support of F supp F
supp F=
All the KH(X)'s subsets that support are compact KC(X)
8
(Theorem)
When X is locally compact Hausdorff, arbitrary element F of KC(X) is globally expressed.
9
Manifold is locally compact Hausdorff.
[References]
Distance Theory / Tokyo May 5, 2004
Actual Language and Imaginary Language / Tokyo September 23, 2004
Distance / Distance Theory Algebraically Supplemented / Tokyo Ocotber 26, 2007
Finsler Manifold and Distance / Energy Distance Theory / Tokyo November 7, 2008
Distance of Word / Complex Manifold Deformation Theory / Tokyo November 30, 2008
Tokyo May 30, 2009
Sekinan Research Field of Language
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